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A link is called quasipositive if it has a special braid diagram, namely a product of conjugates of the positive standard generators of the braid group. If this product only contains words of the form $\sigma_{i,j} = (\sigma_i \cdots \sigma_{j-2})\sigma_{j-1} ( \sigma_i \cdots \sigma_{j-2})^{-1}$ then we call the link strongly quasipositive. These notions were introduced by Rudolph.

Is every quasipositive knot strongly quasipositive?

On pp.102 of Rudolph's Quasipositivity and new knot invariants, he gives an example of a quasipositive link (of 3 components) which is not strongly quasipositive. Is there an example of a knot which is quasipositive but not strongly quasipositive?

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up vote 10 down vote accepted

As pointed out by Hedden, Livingston showed that strongly quasipositive knots have $g(K) = g_4(K) = \tau(K)$, where $g_4$ is the smooth slice genus and $\tau$ is the Ozsváth-Szabó concordance invariant, so this is an easy way to show that some quasipositive knots aren't strongly quasipositive. Baader gave a list of knots which are quasipositive but not positive up to 10 crossings, and among these the criterion $g = g_4$ is actually equivalent to strong quasipositivity, so for example $8_{20}$ (with $g=1$ but $g_4=0$) is quasipositive but not strongly quasipositive.

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Thank you! This is exactly what I was seeking. – Aru Ray Jun 4 '13 at 21:25

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